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Numerical Instability of the Akhmediev Breather and a Finite-Gap Model of It

  • P. G. Grinevich
  • P. M. SantiniEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 273)

Abstract

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in Nature. In this paper we study the numerical instabilities of the Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.

Keywords

Akhmediev breather Rogue waves Split-step Fourier method 

Notes

Acknowledgments

Two visits of P. G. Grinevich to Roma were supported by the University of Roma “La Sapienza”, and by the INFN, Sezione di Roma. P. G. Grinevich and P. M. Santini acknowledge the warm hospitality and the local support of CIC, Cuernavaca, Mexico, in December 2016. P.G. Grinevich was also partially supported by RFBR grant 17-51-150001. We acknowledge useful discussions with F. Briscese, F. Calogero, C. Conti, E. DelRe, A. Degasperis, A. Gelash, I. Krichever, A. Its, S. Lombardo, A. Mikhailov, D. Pierangeli, M. Sommacal and V. Zakharov.

References

  1. 1.
    Ablowitz, M., Herbst, B.: On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math. 339–351 (1990)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ablowitz, M.J., Schober, C.M., Herbst, B.M.: Numerical chaos, roundoff errors and homoclinic manifolds. Phys. Rev. Lett. 71, 2683 (1993)CrossRefGoogle Scholar
  3. 3.
    Ablowitz, M.J., Hammack, J., Henderson, D., Schober, C.M.: Long-time dynamics of the modulational instability of deep water waves. Physica D 152153, 416–433 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Akhmediev, N.N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys 69(2), 1089–1093 (1986)CrossRefGoogle Scholar
  5. 5.
    Akhmediev, N.N., Eleonskii, V.M., Kulagin, N.E.: Generation of periodic sequence of picosecond pulses in an optical fibre: exact solutions. J. Exp. Theor. Phys. 61, 894–899 (1985)Google Scholar
  6. 6.
    Akhmediev, N.N., Eleonskii, V.M., Kulagin, N.E.: Exact first order solutions of the Nonlinear Schödinger equation. Theor. Math. Phys. 72(2), 809–818 (1987)CrossRefGoogle Scholar
  7. 7.
    Akhmediev, N.N.: Nonlinear physics: Déjà vu in optics. Nature (London) 413, 267–268 (2001)CrossRefGoogle Scholar
  8. 8.
    Agrawal, G.P.: Nonlinear Fiber Optics, 3rd edn. Academic Press, San Diego, USA (2001). ISBN 0-12-045143-3Google Scholar
  9. 9.
    Baronio, F., Degasperis, A., Conforti, M., Wabnitz, S.: Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109(4), 44102 (2012)CrossRefGoogle Scholar
  10. 10.
    Belokolos, E.D., Bobenko, A.I., Enolski, V.Z., Its, A.R., Matveev, V.B.: Algebro-geometric Approach in the Theory of Integrable Equations. Springer Series in Nonlinear Dynamics. Springer, Berlin (1994)Google Scholar
  11. 11.
    Benjamin, T.B., Feir, J.E.: The disintegration of wave trains on deep water. Part I Theory. J. Fluid Mech. 27(3), 417–430 (1967)CrossRefGoogle Scholar
  12. 12.
    Biondini, G., Kovacic, G.: Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys. 55, 031506 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Biondini, G., Li, S., Mantzavinos, D.: Oscillation structure of localized perturbations in modulationally unstable media. Phys. Rev. E 94, 060201(R) (2016)CrossRefGoogle Scholar
  14. 14.
    Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80, 033610 (2009)CrossRefGoogle Scholar
  15. 15.
    Bortolozzo, U., Montina, A., Arecchi, F.T., Huignard, J.P., Residori, S.: Spatiotemporal pulses in a liquid crystal optical oscillator. Phys. Rev. Lett. 99(2), 3–6 (2007)CrossRefGoogle Scholar
  16. 16.
    Calini, A., Ercolani, N.M., McLaughlin, D.W., Schober, C.M.: Mel’nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. Physica D 89, 227–260 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Calini, A., Schober, C.M.: Homoclinic chaos increases the likelihood of rogue wave formation. Phys. Lett. A 298(5–6), 335–349 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Calini, A., Schober, C.M.: Dynamical criteria for rogue waves in nonlinear Schrödinger models. Nonlinearity 25, R99–R116 (2012)CrossRefGoogle Scholar
  19. 19.
    Degasperis, A., Lombardo, S.: Integrability in action: solitons, instability and rogue waves. In: Onorato M., Resitori S., Baronio F. (eds.) Rogue and Shock Waves in Nonlinear Dispersive Media. Lecture Notes in Physics. http://www.springer.com/us/book/9783319392127 (2016)CrossRefGoogle Scholar
  20. 20.
    Dubard, P., Gaillard, P., Klein, C., Matveev, V.B.: On multi-rogue waves solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Spec. Top. 185, 247–258 (2010)CrossRefGoogle Scholar
  21. 21.
    Dysthe, K.B., Trulsen, K.: Note on breather type solutions of the NLS as models for freak-waves. Physica Scripta. T82, 48–52 (1999)CrossRefGoogle Scholar
  22. 22.
  23. 23.
    Grinevich, P.G., Santini, P.M.: The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1. Nonlinearity, 31(11), 5258–5308 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Grinevich, P.G., Santini, P.M.: The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes. Physics Letters A. 382, 973–979 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Grinevich P.G., Santini P.M.: The finite gap method and the periodic NLS Cauchy problem of the anomalous waves, for a finite number of unstable modes. arXiv:1810.09247 (2018)
  26. 26.
    Henderson, K.L., Peregrine, D.H., Dold, J.W.: Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equtation. Wave Motion 29, 341–361 (1999)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hirota, R.: Direct Methods for Finding Exact Solutions of Nonlinear Evolution Equations. Lecture Notes in Mathematics, vol. 515. Springer, New York (1976)CrossRefGoogle Scholar
  28. 28.
    Its, A.R., Kotljarov, V.P.: Explicit formulas for solutions of a nonlinear Schrödinger equation. Dokl. Akad. Nauk Ukrain. SSR Ser. A 1051:965–968 (1976)Google Scholar
  29. 29.
    Its, A.R., Rybin, A.V., Sall, M.A.: Exact integration of nonlinear Schrödinger equation. Theor. Math. Phys. 74, 20–32 (1988)CrossRefGoogle Scholar
  30. 30.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits. Phys. Rew. E 85, 066601 (2012)CrossRefGoogle Scholar
  31. 31.
    Kharif C. and Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B/ Fluids J. Mech. 22, 603–634 (2004)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kharif, C., Pelinovsky, E.: Focusing of nonlinear wave groups in deep water. JETP Lett. 73, 170–175 (2001)CrossRefGoogle Scholar
  33. 33.
    Kimmoun, O., Hsu, H.C., Branger, H., Li, M.S., Chen, Y.Y., Kharif, C., Onorato, M., Kelleher, E.J.R., Kibler, B., Akhmediev, N., Chabchoub, A.: Modulation instability and phase-shifted Fermi-Pasta-Ulam recurrence. Sci. Rep. 6, 28516 (2016)CrossRefGoogle Scholar
  34. 34.
    Krichever, I.M.: Methods of algebraic Geometry in the theory on nonlinear equations. Russ. Math. Surv. 32, 185–213 (1977)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Krichever, I.M.: Spectral theory of two-dimensional periodic operators and its applications. Russ. Math. Surv. 44(2), 145–225 (1989)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Krichever, I.M.: Perturbation theory in periodic problems for two-dimensional integrable systems. Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 9(2), 1–103 (1992)Google Scholar
  37. 37.
    Kuznetsov, E.A.: Solitons in a parametrically unstable plasma. Sov. Phys. Dokl. 22, 507–508 (1977)Google Scholar
  38. 38.
    Kuznetsov, E.A.: Fermi-Pasta-Ulam recurrence and modulation instability. JETP Lett. 105(2), 125–129 (2017)CrossRefGoogle Scholar
  39. 39.
    Lake, B.M., Yuen, H.C., Rungaldier, H., Ferguson, W.E.: Nonlinear deep-water waves: theory and experiment. Part 2 Evolution of a continuous wave train. J. Fluid Mech. 83(2), 49–74 (1977)CrossRefGoogle Scholar
  40. 40.
    Ma, Y.-C.: The perturbed plane wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60, 43–58 (1979)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)CrossRefGoogle Scholar
  42. 42.
    Novikov, S.P.: The periodic problem for the Korteweg-de Vries equation. Funct. Anal. Appl. 8(3), 236–246 (1974)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Osborne, A., Onorato, M., Serio, M.: The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys. Lett. A 275, 386–393 (2000)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. Ser. B 25, 16–43 (1983)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Pierangeli, D., Di Mei, F., Conti, C., Agranat, A.J., DelRe, E.: Spatial rogue waves in photorefractive ferroelectrics. PRL 115, 093901 (2015)CrossRefGoogle Scholar
  47. 47.
    Salasnich, L., Parola, A., Reatto, L.: Modulational instability and complex dynamics of confined matter-wave solitons. Phys. Rev. Lett. 91, 080405 (2003)CrossRefGoogle Scholar
  48. 48.
    Smirnov, A.O.: Periodic two-phase rogue waves. Math. Not. 94(6), 897–907 (2013)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)CrossRefGoogle Scholar
  50. 50.
    Stokes, G.: On the theory of oscillatory waves. In: Transactions of the Cambridge Philosophical Society, vol. VIII, 197229, and Supplement 314326 (1847)Google Scholar
  51. 51.
    Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation (Self Focusing and Wave Collapse). Springer, Berlin (1999)zbMATHGoogle Scholar
  52. 52.
    Vespalov, V.I., Talanov, V.I.: Filamentary structure of light beams in nonlinear liquids. JETP Lett. 3(12), 307 (1966)Google Scholar
  53. 53.
    Taniuti, T., Washimi, H.: Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma. Phys. Rev. Lett. 21, 209–212 (1968)CrossRefGoogle Scholar
  54. 54.
    Weideman, J.A.C., Herbst, B.M.: Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23, 485–507 (1986)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Taha, T.R., Xu, X.: Parallel split-step fourier methods for the coupled nonlinear Schrödinger type equations. J Supercomput. 5, 5–23 (2005)CrossRefGoogle Scholar
  56. 56.
    Van Simaeys, G., Emplit, P., Haelterman, M.: Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave. Phys. Rev. Lett. 87, 033902 (2001)CrossRefGoogle Scholar
  57. 57.
    Yuen, H.C., Ferguson, W.E.: Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids 21, 1275 (1978)CrossRefGoogle Scholar
  58. 58.
    Yuen, H., Lake, B.: Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229 (1982)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Zakharov, V.E.: Stability of period waves of finite amplitude on surface of a deep fluid. JAMTP 9(2), 190–194 (1968)Google Scholar
  60. 60.
    Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34(1), 62–69 (1972)MathSciNetGoogle Scholar
  61. 61.
    Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering transform I. Funct. Anal. Appl. 8, 226–235 (1974)CrossRefGoogle Scholar
  62. 62.
    Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Sov. Phys. JETP 47, 1017–27 (1978)Google Scholar
  63. 63.
    Zakharov, V.E., Gelash, A.A.: On the nonlinear stage of Modulation Instability. PRL 111, 054101 (2013)CrossRefGoogle Scholar
  64. 64.
    Zakharov, V., Ostrovsky, L.: Modulation instability: the beginning. Phys. D Nonlinear Phenom. 238(5), 540–548 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.L.D. Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyMoscow RegionRussia
  4. 4.Dipartimento di FisicaUniversità di Roma “La Sapienza”, and Istituto Nazionale di Fisica NucleareRomaItaly

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